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Let X be a space and 2^X(C(X);K(X);CK(X)) denote the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, subcontinua)of X with the Vietoris topology. We investigate the relationships between the space X and its hyperspaces concerning the properties of connectedness im kleinen. We obtained the following : Let X be a locally compact Hausdor® space. Let x 2 X.
Then the following statements are equivalent: (1) X is connected im kleinen at x.
(2) 2^X is arcwise connected im kleinen at {x}. (3) K(X) is arcwise connected im kleinen at fxg. (4) C_K(X) is arcwise connected im kleinen at fxg. (5) C(X) is arcwise connected im kleinen at {x}.
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참고문헌 (8)
참고문헌 신청
A.K. Misra / 1990 / C-supersets, piecewise order-arcs and local arcwise connectedness in hy-perspaces / Q & A in General Topology 8 : 467 ~ 485
Charles Dorsett / 1979 / Local connectedness, connectedness im Kleinen and other properties of hyperspaces of R0 Spaces / Math. Vesnik 3 : 113 ~ 123
E. Michael / 1951 / Topology on spaces of subsets / Trans. Amer. Math. Soc 71 : 152 ~ 182
Jack T. Goodykoontz Jr / 1974 / Connectedness im Kleinen and local connectedness in 2X and C(X) / Pacific J. of Math 53 : 387 ~ 397
Jack T. Goodykoontz Jr. / 1998 / Local properties of hyperspaces / Proceedings 23 : 183 ~ 200
Charles Dorsett / 1979 / Local connectedness, connectedness im Kleinen and other properties of hyperspaces of R0 Spaces / Math. Vesnik 3 : 113 ~ 123
E. Michael / 1951 / Topology on spaces of subsets / Trans. Amer. Math. Soc 71 : 152 ~ 182
Jack T. Goodykoontz Jr / 1974 / Connectedness im Kleinen and local connectedness in 2X and C(X) / Pacific J. of Math 53 : 387 ~ 397
Jack T. Goodykoontz Jr. / 1998 / Local properties of hyperspaces / Proceedings 23 : 183 ~ 200
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